(x+1)(2x+5)=1

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Solution for (x+1)(2x+5)=1 equation: 


Simplifying
(x + 1)(2x + 5) = 1

Reorder the terms:
(1 + x)(2x + 5) = 1

Reorder the terms:
(1 + x)(5 + 2x) = 1

Multiply (1 + x) * (5 + 2x)
(1(5 + 2x) + x(5 + 2x)) = 1
((5 * 1 + 2x * 1) + x(5 + 2x)) = 1
((5 + 2x) + x(5 + 2x)) = 1
(5 + 2x + (5 * x + 2x * x)) = 1
(5 + 2x + (5x + 2x2)) = 1

Combine like terms: 2x + 5x = 7x
(5 + 7x + 2x2) = 1

Solving
5 + 7x + 2x2 = 1

Solving for variable 'x'.

Reorder the terms:
5 + -1 + 7x + 2x2 = 1 + -1

Combine like terms: 5 + -1 = 4
4 + 7x + 2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
4 + 7x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
2 + 3.5x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 3.5x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 3.5x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 3.5x + x2 = 0 + -2
3.5x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
3.5x + x2 = -2

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = -2 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = -2 + 3.0625

Combine like terms: -2 + 3.0625 = 1.0625
3.0625 + 3.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + 1.75) equal to 1.030776406 and -1.030776406.


Subproblem 1
x + 1.75 = 1.030776406

Simplifying
x + 1.75 = 1.030776406

Reorder the terms:
1.75 + x = 1.030776406

Solving
1.75 + x = 1.030776406

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = 1.030776406 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = 1.030776406 + -1.75
x = 1.030776406 + -1.75

Combine like terms: 1.030776406 + -1.75 = -0.719223594
x = -0.719223594

Simplifying
x = -0.719223594


Subproblem 2
x + 1.75 = -1.030776406

Simplifying
x + 1.75 = -1.030776406

Reorder the terms:
1.75 + x = -1.030776406

Solving
1.75 + x = -1.030776406

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = -1.030776406 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = -1.030776406 + -1.75
x = -1.030776406 + -1.75

Combine like terms: -1.030776406 + -1.75 = -2.780776406
x = -2.780776406

Simplifying
x = -2.780776406


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.719223594, -2.780776406}

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